The field equations that are derived from the action considered by Brans and Dicke are of second order for the metric components and the scalar field.
It is possible, however, to construct even more general action functional that produces second order field equations, allowing one to avoid the fatal Ostrogradsky instability which can be present when higher-order theories are considered - so-called Horndeski theory [1]}. Furthermore, it was demonstrated that Horndeski theories are not the most general healthy ST theories: it is possible to construct higher-order theories avoiding the Ostrogradsky instability by being degenerate (so-called DHOST) [2]}, [3]}. It was discovered that the degeneracy of Lagrangian is an essential part of theories with one scalar degree of freedom allowing one to avoid the mentioned problem.
